Methods and apparatus for reducing stick-slip

ABSTRACT

A method and apparatus for estimating the instantaneous rotational speed of a bottom hole assembly at the lower end of a drill string. In one embodiment, a method includes driving the drill string by a drilling mechanism at the upper end of the drill string. A fundamental frequency of stick-slip oscillations suffered by the drill string is estimated. Variations in a drive torque of the drilling mechanism are determined. Known torsional compliance of the drill string is combined with the variations in the drive torque. An output signal representing the instantaneous rotational speed is provided.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority from PCT patent application numberPCT/GB2008/051144, filed 2 Dec. 2008, UK patent application number0907760.3, filed 7 May 2009, and PCT patent application numberPCT/GB2009/051618 filed 30 Nov. 2009. The disclosure of each of thoseapplications is hereby incorporated by reference in its entirety whereappropriate for teachings of additional or alternative details,features, and/or technical background, and priority is asserted.

FIELD OF THE INVENTION

The present invention relates to a method of damping stick-sliposcillations in a drill string, to a method of drilling a borehole, to amethod of estimating the instantaneous rotational speed of a bottom holeassembly, to a drilling mechanism for use in drilling a borehole, to anelectronic controller for use with a drilling mechanism, and to a methodof upgrading a drilling mechanism on a drilling rig.

BACKGROUND

Drilling an oil and/or gas well involves creation of a borehole ofconsiderable length, often up to several kilometres vertically and/orhorizontally by the time production begins. A drill string comprises adrill bit at its lower end and lengths of drill pipe that are screwedtogether. The whole drill string is turned by a drilling mechanism atthe surface, which in turn rotates the bit to extend the borehole. Thedrilling mechanism is typically a top drive or rotary table, each ofwhich is essentially a heavy flywheel connected to the top of the drillstring.

The drill string is an extremely slender structure relative to thelength of the borehole, and during drilling the string is twistedseveral turns because of torque-on-bit between about 500 and 10,000 Nm.The drill string also displays a complicated dynamic behaviourcomprising axial, lateral and torsional vibrations. Simultaneousmeasurements of drilling rotation at the surface and at the bit haverevealed that the drill string often behaves as a torsional pendulumi.e. the top of the drill string rotates with a constant angularvelocity, whereas the drill bit performs a rotation with varying angularvelocity comprising a constant part and a superimposed torsionalvibration. In extreme cases, the torsional part becomes so large thatthe bit periodically comes to a complete standstill, during which thedrill string is torqued-up until the bit suddenly rotates again at anangular velocity that is much higher than the angular velocity measuredat the surface. This phenomenon is known as stick-slip.

Stick-slip has been studied for more than two decades and it isrecognized as a major source of problems, such as excessive bit wear,premature tool failures and poor drilling rate. One reason for this isthe high peak speeds occurring during in the slip phase. The highrotation speeds in turn lead to secondary effects like extreme axial andlateral accelerations and forces.

A large number of papers and articles have addressed the stick-slipproblem. Many papers focus on detecting stick-slip motion and oncontrolling the oscillations by operational means, such as addingfriction reducers to the mud, changing the rotation speed or the weighton bit. Even though these remedies sometimes help, they are eitherinsufficient or they represent a high extra costs.

A few papers also recommend applying smart control of the top drive todampen and prevent stick-slip oscillations. In IADC/SPE 18049 it wasdemonstrated that torque feed-back from a dedicated string torque sensorcould effectively cure stick-slip oscillations by adjusting the speed inresponse to the measured torque variations. In Jansen J. D. et al.“Active Damping of Self-Excited Torsional Vibrations in Oil Well Drillstrings”, 1995, Journal of Sound and Vibrations, 179(4), 647-668, it wassuggested that the drawback of this approach is the need for a new anddirect measurement of the string torque, which is not already available.U.S. Pat. No. 5,117,926 disclosed that measurement as another type offeedback, based on the motor current (torque) and the speed. This systemhas been commercially available for many years under the trade mark SOFTTORQUE®. The main disadvantage of this system is that it is a cascadecontrol system using a torque feedback in series with the stiff speedcontroller. This increases the risk of instabilities at frequencieshigher than the stick-slip frequency.

IADC/SPE 28324 entitled “Application of High Sampling Rate DownholeMeasurements for Analysis and Cure of Stick-Slip in Drilling” disclosescontrol of a drilling process using driving equipment that includes aPID, a motor, a gear box and rotary table. The PID tries to maintain thedesired rotary speed of the drill string and it is suggested that thePID can be adjusted to prevent stick-slip. However, a simulation resultshows poor damping of stick-slip oscillations and it is concluded in thepaper that PID is too simple a servo-control system to preventstick-slip.

Our co-pending patent application PCT/GB2008/051144 discloses a methodfor damping stick-slip oscillations, the maximum damping taking place ator near a first or fundamental (i.e. lowest frequency) stick-sliposcillation mode. In developing the method we have identified a furtherproblem to be addressed when the drill string is extremely long (greaterthan about 5 km) and the fundamental stick-slip period exceeds about 5or 6 s. Even though the method of our previous patent application isable to cure the fundamental stick-slip oscillation mode in suchstrings, as soon as these oscillations are dampened, the second naturalmode tends to become unstable and grow in amplitude until fullstick-slip is developed at the higher frequency. In certain simulationswe have found that this second mode has a natural frequency which isapproximately three times higher than the fundamental stick-slipfrequency. The higher order stick-slip oscillations are characterised byshort period and large amplitude cyclic variations of the drive torque.Simulations show that the bit rotation speed also in this case variesbetween zero and peak speeds exceeding twice the mean speed.

We have also found through other simulations that the method employed bythe aforementioned SOFT TORQUE® system suffers from the same problem.Neither method is able to inhibit both the first and second modestick-slip oscillations.

SUMMARY

Embodiments of the present disclosure are based on the insight that a PIor PID controller can in fact be used to obtain significant damping ofstick-slip oscillations by the drilling mechanism. In particular we haverealised that a PI or PID controller can be tuned to ensure efficientdamping torsional wave energy at and/or near the fundamental mode ofstick-slip frequency. A further insight on which certain embodiments arebased is that both the fundamental and one or more higher mode (e.g.second natural mode and greater) oscillation can also be damped byreducing the effective inertia of the drilling mechanism, which may beachieved in several different ways. One way is by further adjustment ofthe PI or PID controller. Another way is by changing the drillingmechanism to a higher gear. In some embodiments the fundamental and oneor more higher mode may be damped selectively either by a computerdecision in advance (e.g. using predictions based on string geometry).In other embodiments the damping may be selectively activated bymonitoring the period of the fundamental mode and applying the methodwhen the period of the fundamental exceeds a certain threshold.

In contrast to some earlier systems, various embodiments disclosedherein are passive in the sense that neither string torque nor drivetorque is needed in a feed-back loop. Accordingly, damping can beachieved without the need for additional sensors to measure stringtorque, that otherwise increases complexity and cost.

Some embodiments of the invention are based upon the insight that it ispossible to estimate the instantaneous rotational speed of the bit(ignoring any contribution from an optional mud motor) and to make thisinformation available to other control processes on the rig and/or tothe driller via a console. By repeating the method, a substantiallyreal-time estimation of bit speed can be provided. Provision of thisdata may help a driller and/or other automated drilling control processdetermine whether the PI tuning disclosed herein would improve drillingperformance e.g. by reducing stick-slip.

According to some embodiments of the invention there is provided amethod of damping stick-slip oscillations in a drill string, whichmethod comprises the steps of:

(a) damping said stick-slip oscillations using a drilling mechanism atthe top of said drill string; and

(b) controlling the speed of rotation of said drilling mechanism using aPI controller;

characterised by the step of

(c) tuning said PI controller so that said drilling mechanism absorbsmost torsional energy from said drill string at a frequency that is ator near a frequency of said stick-slip oscillations, and/or at or near afundamental frequency and at least one higher frequency mode of saidstick-slip oscillations. The drilling mechanism may comprise a top driveor a rotary table for example. It is to be noted that the PI controllermay be tuned once (for example upon encountering stick-slip for thefirst time, or in advance of drilling) and upon subsequent occurrencesof stick-slip the PI controller may be used again without beingre-tuned. Another possibility is for the PI controller to be re-tunedeach time stick-slip is encountered, or even periodically during astick-slip phase of drilling. In one embodiment, the PI controller istuned before it is used to control the drilling mechanism to dampstick-slip oscillations. For example, the controller may be tuned uponencountering stick-slip oscillations or it may be performed periodicallyduring drilling of the borehole as the drill string length increases.One possibility is for the tuning to take place as each 30 m section ofdrill pipe is added to the drill string.

In certain embodiments, the PI controller may adjusted to damp both afundamental frequency and one or more higher mode stick-sliposcillations; the options for such tuning include: tuning in advance ofdrilling (for example on the basis of predictions using string geometry,or simply as a precaution against higher mode oscillations whether theyare expected or not), tuning on encountering a fundamental mode (whetheror not higher modes are expected) or tuning on encountering higher modestick-slip oscillations.

In some embodiments said stick-slip oscillations comprise torsionalwaves propagating along said drill string, and step (c) comprisesadjusting an I-term of said PI controller to be dependent on anapproximate period of said fundamental frequency of said stick sliposcillations and on an effective inertia of said drilling mechanism,whereby said drilling mechanism has a frequency dependent reflectioncoefficient of said torsional waves, which reflection coefficient issubstantially at a minimum at or near said fundamental frequency ofstick-slip oscillations. It is to be noted that it is not essential forthe peak absorption frequency of the drilling mechanism to match exactlythe fundamental frequency of the stick-slip oscillations. Due to the waythe PI controller is tuned, the drilling mechanism has a bandwidth offrequency absorption that is of a sufficient width (e.g. ˜0.4 Hz) andmagnitude (e.g. less than 85% reflection) so that damping is stilleffective even if the two frequencies are not exactly matched. Thisrepresents a significant advantage of the method. Typically, thefundamental frequency of stick-slip oscillations encountered in practicelies in the range 0.1 Hz (period 10 s) to 0.5 Hz (period 2 s) and thepeak absorption frequency caused by the PI controller may be within 50%of the fundamental frequency.

In some embodiments the lowest point of the frequency-reflectioncoefficient curve has a value between about 50% (0.5) and 90% (0.9). Ithas been found that reflection coefficients any higher than about 90%can make the drilling mechanism too “stiff” and reduce the chance ofsuccessfully damping the stick-slip oscillations. On the other hand, ithas been found that a reflection coefficient of any lower than about 50%makes the drilling mechanism too “soft” and drilling performance can beimpaired since the drilling mechanism responds to much smaller changesin drill string torque resulting in high speed variations.

The absorption bandwidth is inversely proportional to the effectiveinertia J of the drilling mechanism. Therefore as the effective inertiaof a drilling mechanism increases, it is preferable although notessential, that the approximate stick-slip period is estimated ormeasured more accurately to ensure that the frequency of greatestdamping is real stick-slip frequency.

In some embodiments, the method further comprises the step of adjustingsaid I-term according to I=ω_(s) ²J where ω_(s) is an approximate orestimated angular frequency of said stick-slip oscillations and J is theeffective inertia of said drilling mechanism. ω_(s) could of course beexpressed in terms of other parameters in this formula, such as theperiod or frequency.

In certain embodiments, said effective inertia comprises the totalmechanical inertia of said drilling mechanism at an output shaftthereof. This has been found useful for damping predominantly only afundamental mode of stick-slip oscillation, although higher modes aredamped to some extent.

In other embodiments, the method further comprises the step of reducingan effective inertia of said drilling mechanism, whereby a dampingeffect of said drilling mechanism is increased for frequencies abovesaid fundamental frequency. This is a significant optional step of themethod that enables one or more higher mode oscillations to be damped(and in some embodiments cured altogether) at the same time as dampingthe fundamental mode. This possibility is particularly important forlong drill strings (typically over about 5 km in length), where highermode oscillations are likely to be problematic. Reduction of effectiveinertia may be applied continuously (whether or not higher modestick-slip is expected) or selectively either upon detection of afundamental mode of period greater than a certain threshold (e.g. fiveseconds), or in response to detection of one or more higher mode whilstdrilling. Furthermore, the quantity of inertia reduction may be adjustedto change the amount of damping at higher frequencies.

In some embodiments the step of reducing said effective inertiacomprises the step of tuning said PI controller with an additionaltorque term that is proportional to the angular acceleration of saiddrilling mechanism. Since the angular acceleration is readily derivedfrom the angular speed of the drilling mechanism, this makes the methodvery easy to implement in a computer operated speed controller (forexample a controller implemented in a PLC).

In certain aspects, the method further comprises the step of multiplyingsaid angular acceleration by a compensation inertia (J_(c)), whichcompensation inertia (J_(c)) is adjustable so as to control the amountof the reduction of the effective inertia of said drilling mechanism.The compensation inertia may be a relatively static value (e.g. set by adriller via a console) or a dynamic value (e.g. adjusted in real timeaccording to drilling conditions). Typically the compensation inertia(J_(c)) may be adjusted so as to reduce said effective inertia bybetween 0 and 80%.

In some embodiments, the method further comprises the step of adjustingsaid I-term of said PI controller according to I=ω_(s) ²J, where ω_(s)is an approximate or estimated angular frequency of said stick-sliposcillations and J is the reduced effective inertia value of saiddrilling mechanism.

In certain embodiments, said drilling mechanism has a torsional energyabsorption bandwidth for stick-slip oscillations, the size of saidbandwidth obtainable from its full width half maximum, whereby uponreducing the effective inertia of said drilling mechanism the size ofsaid full width half maximum (FWHM) is greater. Use of the FWHM providesa convenient way to compare different absorption bandwidths.

In some embodiments said drilling mechanism has a frequency dependentdamping curve having a point of maximum damping, the method furthercomprising the step of shifting said point of maximum damping to higherfrequencies whereby the damping effect of said drilling mechanism on atleast some higher frequencies is increased and damping of saidfundamental frequency is reduced. This is referred to herein asde-tuning, and optionally, is performed if higher mode stick-sliposcillations are not reduced or cured by the inertia compensationmethod.

In some aspects, said step of shifting comprises determining an I-termof said PI controller as I=ω_(s) ²J, in which a period value ω_(s) isgreater than said approximate period of said fundamental frequency,whereby said frequency dependent damping curve is shifted toward higherfrequencies and damping of at least one higher mode of oscillation isincreased above the amount of damping obtainable when using saidapproximate period to determine said I-term. The period value may be 40%greater than said approximate period.

In some embodiments, the method further comprises the step of furtherreducing said effective inertia of said drilling mechanism whenperforming said shifting step, whereby narrowing of an absorptionbandwidth of said damping curve is inhibited. In certain aspects thismay be achieved by reducing said effective inertia and increasing saidperiod value by the same factor.

In other embodiments, the step of reducing said effective inertiacomprises changing into a higher gear of said drilling mechanism.Instead of achieving an effective inertia reduction through a speedcontroller, a similar effect may be achieved by changing into a highergear (assuming the drilling mechanism has more than one gear). In thisway it is envisaged that the PI controller could be tuned to damppredominantly the fundamental stick-slip frequency and, if and when oneor more higher mode oscillation is encountered, the drilling mechanismmay be shifted into a higher gear to increase damping at higherfrequencies.

In other embodiments, the method further comprises the steps ofmonitoring said drilling mechanism for occurrence of one or more highermode of oscillation, and when detected, performing any of the highermode damping steps set out above in order to damp said one or morehigher mode of oscillation. The monitoring may be performed by computerobservation of the speed of rotation of the drilling mechanism forexample.

In other aspects the method further comprises the steps of monitoring aperiod of said fundamental frequency, comparing said period against aperiod threshold and, if said period exceeds said period threshold,performing any of the higher mode damping steps set out above to dampsaid one or more higher mode of oscillation. One example of the periodthreshold is five seconds. Once the fundamental stick slip periodincreases beyond that, the effective inertia is reduced to counter-actany higher mode oscillations. In some embodiments, above said periodthreshold, said effective inertia is reduced as said period increases.For example, the effective inertia may be reduced as a function of themonitored period. In one particular example, the effective inertia isreduced linearly from 100% to 25% of its full value as the monitoredperiod increases between about five seconds and eight seconds.

In some embodiments, the PI controller may comprise a PID controller inwhich the derivative term is not used in implementation of effectiveinertia reduction. For example a standard digital PID controller may beadapted (e.g. by adjustment of low-level source code) to implementeffective inertia reduction.

In other embodiments, the method further comprises the step of measuringsaid approximate period of stick-slip oscillations for use in adjustingsaid I-term. In certain embodiments this measurement may be performedautomatically by a PLC for example. In that case, the approximate periodmay be determined using drill string geometry or it may be determined bycomputer observation of drive torque. Another possibility is for theapproximate period to be estimated by the driller, for example by timingwith a stop-watch torque oscillations shown on the driller's console, orby simply listening to changes in pitch of the motor(s) of the drillingmechanism and timing the period that way. The driller may input theapproximate stick-slip period into a console to be processed by a PLC totune the I-term of the PI controller.

In some embodiments, the method further comprises the step of adjustinga P-term of said PI controller to be the same order of magnitude as thecharacteristic impedance ζ of said drill string. In this way thereflection coefficient of the drilling mechanism can be reduced further,increasing the damping effect.

In other embodiments, the method further comprises the step of adjustingsaid P-term such that said reflection coefficient does not vanishcompletely whereby a fundamental mode of said stick slip oscillations isinhibited from splitting into two new modes with different frequencies.

In some embodiments, the method further comprises the step of adjustingsaid P-term as P=ζ/a where a is a mobility factor that permitsadjustment of said P-term during drilling, whereby energy absorption ofsaid stick-slip oscillations by said drilling mechanism may be increasedor reduced. The mobility factor may be adjusted automatically by acontroller (e.g. PLC) and/or may be adjusted manually by the driller. Inthis way the softness of the drilling mechanism can be adjusted toachieve a balance between damping stick-slip oscillations and drillingperformance.

In some aspects the method further comprises the step of increasing saidmobility factor if the magnitude of said stick-slip oscillations do notsubstantially disappear or reduce. In this way the softness of thedrilling mechanism is increased (i.e. is made more responsive to smallertorque variations).

In other aspects the method further comprises the step of reducing saidmobility factor once the magnitude of said stick-slip oscillations hassubstantially disappeared or reduced, whereby drilling efficiency isincreased without re-appearance or increase in magnitude of saidstick-slip oscillations. In this way the softness of the drillingmechanism is reduced (i.e. is made less responsive to smaller torquevariations).

In some embodiments, said PI controller is separate from a drillingmechanism speed controller, the method further comprising the step ofbypassing said drilling mechanism speed controller with said PIcontroller during damping of said stick-slip oscillations. The PIcontroller may be provided on a drilling rig separate from the drillingmechanism, either on a new rig or as an upgrade to an existing rig inthe field. In use, when stick-slip oscillations occur, the PLC mayoverride the dedicated speed controller of the drilling mechanism(either automatically or under control of the driller) to control it asset out above.

In other embodiments, said drilling mechanism comprises said PIcontroller, the method further comprising the steps of tuning said PIcontroller when said stick-slip oscillations occur, and leaving said PIcontroller untuned otherwise. In such embodiments the PI controller maybe part of the dedicated speed controller in a drilling mechanism suchas a top drive. The PI controller may be provided as software installedon a PLC or other computer control mechanism at point of manufacture. Inuse, the PI controller is used continuously but may only need to betuned as described above when stick-slip oscillations occur. This tuningmay be activated automatically be remote drilling control software (e.g.a driller's console on or off site) and/or may be controlled by thedriller using a driller's console.

In some embodiments, the method further comprises the step of estimatingthe instantaneous rotational speed of a bottom hole assembly at thelower end of said drill string by combining a known torsional complianceof said drill string with variations in a drive torque of said drillingmechanism. This is a particularly useful optional feature of someembodiments of the invention and the output may be displayed on adriller's console or otherwise to help the driller to visualise what ishappening downhole.

In other embodiments, variations in drive torque are expressed only at afundamental frequency of said stick-slip oscillations, whereby saidestimating step is simplified such that it may be implemented by a PLCand performed in real time. The drive torque variations comprise afrequency spectrum which makes the drive torque signal difficult toanalyse. We have realised that it is sufficient only to analyse thefundamental frequency component of the drive torque variations and thatthis enables the analysis to be performed in real-time on a PLC forexample.

In some embodiments, said estimating step comprises band pass filteringa drive torque signal with a band pass filter centred on an approximatefrequency of said stick-slip oscillations. This helps to remove most ofthe higher and lower frequencies in the torque signal. The approximatefrequency may be determined as described above.

In certain aspects, said estimate of instantaneous rotational speedcomprises determining a downhole speed using a total static drill stringcompliance and a phase parameter, and determining the sum of (i) a lowpass filtered signal representing a speed of rotation of said drillingmechanism and (ii) said downhole speed.

In other embodiments, the method further comprises the step ofdetermining said estimate periodically and outputting said estimate on adriller's console whereby a driller is provided with a substantiallyreal-time estimate of the instantaneous rotational speed of said bottomhole assembly.

In some embodiments, the method further comprises the step ofdetermining a stick-slip severity as the ratio of dynamic downhole speedamplitude over the mean rotational speed of said drilling mechanism,which stick-slip severity is useable to provide an output signalindicating the severity of stick-slip at that point in time.

According to some embodiments of the invention there is provided amethod of drilling a borehole, which method comprises the steps of:

(a) rotating a drill string with a drilling mechanism so as to rotate adrill bit at a lower end of said drill string whereby the earth'ssurface is penetrated; and

(b) in response to detection of stick-slip oscillations of said drillstring using a PI controller to control said drilling mechanism, whichPI controller has been tuned by a method according to any of claims 1 to27. It is to be noted that the PI controller may be tuned once (forexample upon encountering stick-slip for the first time) and uponsubsequent occurrences of stick-slip the PI controller may be usedwithout re-tuning. Of course, another possibility is for the PIcontroller to be re-tuned each time stick-slip is encountered, or evenas stick-slip is ongoing. The PI tuning method may therefore be usedselectively during drilling to counter stick-slip oscillations. At othertimes the PI controller may be left untuned so that a speed controllerof the drilling mechanism has a standard stiff behaviour (i.e. with areflection coefficient approximately equal to 1).

According to yet another embodiment of the invention there is provided amethod of estimating the instantaneous rotational speed of a bottom holeassembly at the lower end of a drill string, which method comprises thesteps of combining a known torsional compliance of said drill stringwith variations in a drive torque of said drilling mechanism. Such amethod may be performed either on or off site, either during drilling orafter drilling a section of the borehole. Such a method provides adrilling analysis tool to determine if the PI controller tuning aspectof embodiments of the invention would improve drilling performance.Accordingly, software to perform this method may be provided separatelyfrom software to perform the tuning method. The rotational speedestimating software may be provided in the controller of a new drillingmechanism (i.e. included a point of manufacture), as an upgrade to anexisting drilling mechanism (e.g. performed either on site or remotelyusing a satellite connection to a computer system on the drilling rig),or as a computer program product (e.g. on a CD-ROM or as a download froma website) for installation by the rig operator.

In certain aspects, the rotational speed estimating method furthercomprises the estimating steps as set out above.

According to some embodiments of the invention there is provided adrilling mechanism for use in drilling a borehole, which drillingmechanism comprises an electronic controller having a PI controller andmemory storing computer executable instructions that when executed causesaid electronic controller to tune said PI controller according to thetuning steps set out above.

According to other embodiments of the invention there is provided anelectronic controller for use with a drilling mechanism for drilling aborehole, which electronic controller comprises a PI controller andmemory storing computer executable instructions that when executed causesaid electronic controller to tune said PI controller according to thetuning steps set out above. Such an electronic controller is useful forupgrading existing drilling rigs or where it is desirable or necessarythat the electronic controller is separate from the drilling mechanism.

According to further embodiments of the invention there is provided amethod of upgrading a drilling mechanism on a drilling rig, which methodcomprises the steps of uploading computer executable instructions to anelectronic controller on said drilling rig, which electronic controlleris for controlling operation of said drilling mechanism, wherein saidcomputer executable instructions comprise instructions for performing atuning method as set out above. Such an upgrade may be performed onsite, or may be performed remotely using a satellite connection forexample.

According to certain embodiments of the invention there is provided amethod of damping stick-slip oscillations in a drill string, whichmethod comprises the steps of:

(a) damping said stick-slip oscillations using a drilling mechanism atthe top of said drill string; and

(b) controlling the speed of rotation of said drilling mechanism using aPI controller;

characterised by the step of

(c) reducing an effective inertia of said drilling mechanism wherebyboth a fundamental frequency and at least one higher frequency mode(harmonic) of stick-slip oscillation are damped at the same time. Theeffective inertia may be reduced by tuning said PI controller (whichincludes a PID controller) and/or by changing said drilling mechanism toa higher gear.

Certain embodiments of this invention are not limited to any particularindividual feature disclosed here, but include combinations of themdistinguished from the prior art in their structures, functions, and/orresults achieved. Features of various embodiments of the invention havebeen broadly described so that the detailed descriptions that follow maybe better understood, and in order that the contributions of thisinvention to the arts may be better appreciated. There are, of course,additional aspects of the various embodiments of the invention describedbelow and which may be included in the subject matter of the claims.Those skilled in the art who have the benefit of this disclosure, itsteachings, and suggestions will appreciate that the conceptions of thisdisclosure may be used as a creative basis for designing otherstructures, methods and systems for carrying out and practicing thepresent invention. The claims of this disclosure are to be read toinclude any legally equivalent devices or methods which do not departfrom the spirit and scope of the embodiments disclosed herein.

The present disclosure recognizes and addresses the previously mentionedproblems and long felt needs and provides a solution to those problemsand a satisfactory meeting of those needs in its various possibleembodiments and equivalents thereof. To one of skill in this art who hasthe benefits of this disclosure's realizations, teachings, andsuggestions, other purposes and advantages will be appreciated from thefollowing description of certain preferred embodiments, given for thepurpose of disclosure, when taken in conjunction with the accompanyingdrawings. The detail in these descriptions is not intended to thwartthis patent's object to claim this invention no matter how others maylater disguise it by variations in form, changes, or additions offurther improvements.

It will be understood that the various embodiments of the presentinvention may include one, some, or all of the disclosed, described,and/or enumerated improvements and/or technical advantages and/orelements in the claims.

BRIEF DESCRIPTION OF THE FIGURES

For a better understanding of exemplary embodiments of the invention,reference will now be made, by way of example only, to the accompanyingdrawings in which:

FIG. 1 is a schematic side view of a drilling rig using a methodaccording to various embodiments of the present invention;

FIG. 2 is a schematic block diagram of a PLC comprising a speedcontroller according to various embodiments of the present invention;

FIG. 3 is a graph of frequency versus reflection coefficient showing acomparison between a drilling mechanism using a speed controlleraccording to a first embodiment of the present invention and a standardspeed controller;

FIG. 4A′ and 4A″ is a screenshot of a first window available on adriller's console for configuring and controlling a method according tovarious embodiments of the present invention;

FIG. 4B′ and 4B″ is a screenshot of a second window available on adriller's console that illustrates real-time drive torque and anestimate of downhole rotation speed of the bottom hole assembly in FIG.1;

FIGS. 5 and 6 are graphs illustrating results of a computer simulationmodelling of a first method according to various embodiments of thepresent invention;

FIGS. 7 and 8 are graphs illustrating results of a test of a methodaccording to various embodiments of the present invention;

FIG. 9 is a graph of normalised frequencies versus normalized BHAinertia;

FIG. 10 is a graph illustrating the first three torsional oscillationmodes of a drill string;

FIG. 11 is a graph of frequency versus reflection coefficient showing acomparison between a drilling mechanism using: a standard speedcontroller, a speed controller according to a first embodiment thepresent invention, and a speed controller according to a secondembodiment of the present invention;

FIG. 12 is a graph of frequency versus reflection coefficientillustrating a de-tuning aspect of the second embodiment of the presentinvention;

FIG. 13 shows a graph similar to FIG. 11, showing the effect of delayand a low pass filter on a speed controller according to the secondembodiment;

FIG. 14 are graphs illustrating results of a computer simulationmodelling of a second method according to various embodiments of thepresent invention; and

FIG. 15 is a graph of frequency versus reflection coefficient of amethod of damping stick-slip oscillations according to a thirdembodiment of the present invention.

DETAILED DESCRIPTION

Referring to FIG. 1 a drilling rig 10 controls a drilling operationusing a drill string 12 that comprises lengths of drill pipe 14 screwedtogether end to end. The drilling rig 10 may be any sort of oilfield,utility, mining or geothermal drilling rig, including: floating and landrigs, mobile and slant rigs, submersible, semi-submersible, platform,jack-up and drill ship. A typical drill string is between 0 and 5 km ormore in length and has at its lowest part a number of drill collars orheavy weight drill pipe (HWDP). Drill collars are thicker-walled thandrill pipe in order to resist buckling under the compression forces:drill pipe may have an outer diameter of 127 mm and a wall thickness of9 mm, whereas drill collar may have an outer diameter of up to 250 mmand a wall thickness of 85 mm for example.

A bottom hole assembly (BHA) 16 is positioned at the lower end of thedrill string 12. A typical BHA 16 comprises a MWD transmitter 18 (whichmay be for example a wireline telemetry system, a mud pulse telemetrysystem, an electromagnetic telemetry system, an acoustic telemetrysystem, or a wired pipe telemetry system), centralisers 20, adirectional tool 22 (which can be sonde or collar mounted), stabilisers(fixed or variable) and a drill bit 28, which in use is rotated by a topdrive 30 via the drill string 12.

The drilling rig 10 comprises a drilling mechanism 30. The function ofthe drilling mechanism 30 is to rotate the drill string 12 and therebythe drill 28 at the lower end. Presently most drilling rigs use topdrives to rotate the drill string 12 and bit 28 to effect drilling.However, some drilling rigs use a rotary table and embodiments of theinvention are equally applicable to such rigs. Embodiments of theinvention are also equally useful in drilling any kind of borehole e.g.straight, deviated, horizontal or vertical.

A pump 32 is located at the surface and, in use, pumps drilling fluidthrough the drill string 12 through the drill bit 28 and serves to cooland lubricate the bit during drilling, and to return cuttings to thesurface in the annulus formed between the drill string and the wellbore(not shown).

Drilling data and information is displayed on a driller's console 34that comprises a touch screen 36 and user control apparatus e.g.keyboard (not shown) for controlling at least some of the drillingprocess. A digital PLC 38 sends and receives data to and from theconsole 34 and the top drive 30. In particular, a driller is able to seta speed command and a torque limit for the top drive to control thespeed at which the drill bit 28 rotates.

Referring to FIG. 2 the PLC 38 comprises a non-volatile flash memory 40(or other memory, such as a battery backed-up RAM). The memory storescomputer executable instructions that, when executed, perform thefunction of a speed controller 42 for the top drive 30. The speedcontroller 42 comprises a PI controller with anti-windup that functionsas described in greater detail below. In this embodiment the speedcontroller 42 is separate and distinct from the top drive 30. However,it is possible for the functionality of the speed controller asdescribed herein to be provided as part of the in-built dedicated speedcontroller of a top drive. Such in-built functionality may either beprovided at point of manufacture or may be part of a software upgradeperformed on a top drive, either on or off site. In other embodimentsthe PLC may be an analogue PLC.

PI Controller Tuning

The drill string 12 can be regarded as a transmission line for torsionalwaves. A variation of the friction torque at the drill bit 28 orelsewhere along the string generates a torsional wave that is propagatedupwards and is partially reflected at geometric discontinuities. Whenthe transmitted wave reaches the top drive 30, it is partially reflectedback into the drill string 12. For a top drive with a high inertiaand/or a stiff speed controller the reflection is nearly total so thatthat very little energy is absorbed by the top drive.

To quantify the top drive induced damping a complex reflectioncoefficient r for torsional waves at the drill string/top driveinterface may be defined as follows:

$\begin{matrix}{r = \frac{\zeta - Z}{\zeta + Z}} & (1)\end{matrix}$

where ζ is the characteristic impedance for torsional waves and Z is theimpedance of the top drive. The characteristic impedance is proportionalto the cross sectional polar moment of inertia for the pipe, and variesroughly as the 4^(th) power of the pipe diameter. Note that thereflection coefficient is a complex function where, in general, both themagnitude and phase vary with frequency. If the speed control is stiff(i.e. |Z|>>ζ) then the reflection coefficient approaches −1 and nearly100% of the torsional wave energy is reflected back down the drillstring 12 by the top drive 30.

A complex representation of the top drive impedance may be derived asfollows. If the anti wind-up of the speed controller is neglected (whichis a non-linear function that limits torque) the drive torque of the topdrive 30 can be written as:

T _(d) =P(Ω_(set)−Ω)+I∫(Ω_(set)−Ω)dt  (2)

where P and I are respective the proportional and integration factors ofthe speed controller, and Ω is the actual output drive speed (in rad/s)and Ω_(set) is the set point of the drive speed (in rad/s). The drivetorque is actually the sum of motor torques times the gear ratio n_(g)(motor speed/output speed, >1). Notice that speed control here refers tothe output axis of the top drive. It is more common for the speedcontrol to refer to the motor axis; in that case the corresponding P andI values for the motor speed control would then be a factor 1/n_(g) ²lower than above.

Neglecting transmission losses, the equation of motion of the top driveoutput shaft is:

$\begin{matrix}{{J\frac{\Omega}{t}} = {T_{d} - T}} & (3)\end{matrix}$

where J is the effective inertia of the drilling mechanism (includinggear and drive motors) and T is the external torque from the string. Inthis embodiment, the effective inertia is equal to the total mechanicalinertia of the drilling mechanism 30. Combining equations (2) and (3)and applying the Fourier transform gives the following equation ofmotion:

$\begin{matrix}{{\left( {{{\omega}\; J} + P + \frac{I}{\omega}} \right)\Omega} = {{\left( {P + \frac{I}{\omega}} \right)\Omega_{set}} - T}} & (4)\end{matrix}$

For simplicity, the same variable names have been used as in the timebased equations, although Ω, Ω_(set) and T now represent complexamplitudes. The implied time factor is exp(iωt), where i=√{square rootover (−1)} is the imaginary unit and ω=2πf is the angular frequency ofthe top drive 30. If we assume there is no cascade feedback through theset speed (as found in torque feed-back systems), the set speedamplitude vanishes and the equation above simplifies to:

$\begin{matrix}{T = {{- \left( {{{\omega}\; J} + P + \frac{I}{\omega}} \right)}\Omega}} & (5)\end{matrix}$

The negative ratio −T/Ω is called the top end impedance Z of the string:

$\begin{matrix}{Z = {{{\omega}\; J} + P + \frac{I}{\omega}}} & (6)\end{matrix}$

This impedance can easily be generalized to an ideal PID controller, byadding a new term iωD to it, where D is the derivative term of thecontroller. A (normal) positive D-term will increase the effectiveinertia of the top drive (as seen by torsional waves travelling up thedrill string), while a negative factor will reduce it. In practice,because time differentiation of the measured speed is a noise drivingprocess that enhances the high frequency noise, the D-term in a PIDcontroller is normally combined with a low pass filter. This filterintroduces a phase shift that makes the effective impedance morecomplicated and it therefore increases the risk of making instabilitiesat some frequencies, as explained below. Therefore, although a PIDcontroller with a D-term could be used to perform the tuning aspect ofsome embodiments of the invention, it is not recommended. However, inanother aspect of the invention described below, we have found a way toadjust the effective inertia of the drilling mechanism without thisdisadvantage.

Combining equations (1) and (6) gives the following expression for thereflection coefficient, valid for PI type speed controlled top drives:

$\begin{matrix}{r = {- \frac{P - \zeta + { \cdot \left( {{\omega \; J} - \frac{I}{\omega}} \right)}}{P + \zeta + { \cdot \left( {{\omega \; J} - \frac{I}{\omega}} \right)}}}} & (7)\end{matrix}$

Its magnitude has a minimum equal to:

$\begin{matrix}{{r}_{\min} = \frac{{P - \zeta}}{P + \zeta}} & (8)\end{matrix}$

when the imaginary terms vanish, that is, when the angular frequency ofthe top drive 30 equals ω=√{square root over (I/J)}. For standard stiffspeed controllers this frequency is normally higher than the stick-slipfrequency (see FIG. 3 and associated description). However, we havediscovered that adjustment of the I-term of the PI controller alsoadjusts the peak absorption frequency of torsional waves by the topdrive 30. In particular, the I-term can be adjusted so that the maximumenergy absorption of torsional waves occurs at or near the stick-slipfrequency ω_(s) (i.e. when the magnitude of the reflection coefficientis minimum) as follows:

I=ω _(s) ² J  (9)

This realization is significant since, as a first step to achieving gooddamping, the I-term of the PI controller is only dependent on thestick-slip frequency and the effective inertia of the top drive 30.Since the effective inertia is readily determined either in advance ofoperation or from figures quoted by the manufacturer, and since thestick-slip frequency can be readily determined during drilling, thismakes tuning of the PI controller straightforward whilst achieving goodenergy absorption by the top drive 30 of the stick-slip oscillations.

This first step in tuning the speed controller is a good first steptowards effective dampening of stick-slip oscillations. However, thedamping can be further improved. In particular the untuned P-term of thespeed controller is still too high, that is P>>ζ keeping the reflectioncoefficient close to −1. We have discovered that to obtain sufficientdamping of the stick-slip oscillations the P-term of the speedcontroller must be lowered so that it is of the same order of magnitudeas the characteristic impedance ζ. However, we have also discovered thatit is not desirable that the reflection coefficient vanishes completely,because that would radically change the dynamics of the drill string 12and the pendulum mode would split into two new modes, each with adifferent frequency. Furthermore an extremely soft speed controller thatabsorbs nearly all of the incident wave energy will cause very highspeed fluctuations of the top drive 30, in response to variations of thedownhole torque. This can reduce drilling efficiency.

We have discovered that the P-term can be selected as a non-integermultiple of the characteristic impedance ζ of the drill string, whichmay be expressed as P=ζ/a where a is a normalised mobility factor(dimensionless) less than unity, which is operator or computeradjustable within certain limits as described below. Having set theI-term to cause the imaginary part of equation (7) to vanish, settingthe P-term as described causes the minimum of the reflection coefficient(i.e. the peak absorption of energy by the top drive) at the stick-slipfrequency ω_(s) to become:

$\begin{matrix}{{r}_{\min} = \frac{1 - a}{1 + a}} & (10)\end{matrix}$

By permitting adjustment of the mobility factor a, the amount of energyreflected back down the drill string 12 can be controlled, withinlimits. These limits can be set by permitting only a certain range ofvalues for a, such as 0.05 to 0.33. This corresponds to a range for themagnitude of r_(min) from about 0.9 to 0.5. It is believed that thisrange enables the damping to be controlled so that stick-sliposcillations can be inhibited. If the speed controller 42 is muchstiffer than this (i.e. a reflection coefficient greater than about 0.9)we have found that too much of the torsional energy of the stick-sliposcillations is reflected back down the drill-string 12. Furthermore, ifthe speed controller 42 is too soft (i.e. a reflection coefficient lessthan about 0.5) we have found that drilling performance (e.g. in termsof ROP) can be affected.

A standard speed controller is designed to keep the motor speed constantand the true P and I constants refer to the motor axis. A typical drivemotor with a nominal power of 900 kW and a rotor inertia of J_(m)=25kgm² is typically controlled by a motor speed controller of P_(m)=500Nms. The speed controller I-factor is most often given indirectly as theP-factor divided by a time integration constant of typically τ_(i)=0.3s. As an example, assume a drive with one motor connected to the outputshaft with a gear having an inertia J_(g)=250 kgm² and a gear ratio ofn_(g)=5.32. The effective drive inertia (i.e. total mechanical inertia)is then J_(d)=J_(g)+n_(g) ²J_(m)=960 kgm². The effective speedcontroller factors referred to the output shaft is similarly P=n_(g)²P_(m)≈14000 Nms and I=P/τ_(i)47000 Nm. In comparison, thecharacteristic impedance for a typical 5 inch pipe with ζ≈340 Nms, whichis only 2.4% of the real part of the drive impedance.

FIG. 3 is a graph 48 of the magnitude of the reflection coefficient Idversus frequency and shows the difference between a standard stiff speedcontroller (curve 50) and a speed controller tuned according to variousembodiments of the invention (curve 52). The latter is calculated with amobility factor of a=0.25 and an I-term providing maximum damping at 0.2Hz (5 s stick-slip period). At this frequency the reflection is reducedfrom about 0.993 (standard PI controller) to 0.6 (PI controller tuned asabove), which represents a dramatic improvement in the damping by thetop drive at the stick-slip frequency.

It is worth emphasizing the fact that in both cases the reflectioncoefficient stays below 1 but approaches this limit as the frequencyapproaches either zero or infinity. Therefore, the standardPI-controller never provides a negative damping that would otherwiseamplify torsional vibration components. However, the damping is poor faraway from the relatively narrow the absorption band at 1-2 Hz. Incontrast, the tuned PI controller provides a comparatively wideabsorption band with less than 80% reflection between about 0.1 Hz and0.4 Hz. There is even a substantial damping effect remaining (|r|=0.965)at 0.6 Hz, which is three times the stick-slip frequency and close tothe second resonance frequency of the drill string.

The effective inertia J of the drilling mechanism, the characteristicimpedance and the stick-slip frequency ω_(s) change the absorptionbandwidth of the frequency-reflection curve in FIG. 3. In particular,the absorption bandwidth is inversely proportional to the ratioω_(s)J/ζ. For a drilling mechanism with a large effective inertia and/ora slender drill pipe making this ratio larger (e.g. greater than 5), theabsorption bandwidth narrows. In that case, it becomes more important toensure that the estimated stick-slip period is determined moreaccurately (if possible) so that the frequency of maximum damping is asclose as possible to the actual stick-slip frequency.

The reduction in reflection coefficient magnitude and correspondingpositive damping over the entire frequency band is very important and isachieved with only a single PI controller. This is in contrast to otheractive methods that use cascade feed-back loops in series with astandard speed controller, or that rely on some measured parameter suchas drive or string torque to provide a feedback signal to the PLC. Thefilters used in the cascade feed-back functions can be suitable fordamping the fundamental stick-slip oscillations but they can causenegative damping and instabilities at higher frequencies.

In practice, the P-term for the tuned speed controller may be determinedas follows:

$P = {\frac{\zeta}{a} = \frac{{GI}_{p}}{ca}}$

where G is the shear modulus of the drill string (typical value is80×10⁹Nm⁻²), I_(P) is the cross-sectional polar moment of inertia of thedrill string (typical value is 12.2×10⁻⁶ m⁴) and c is the speed oftorsional waves in the drill string (typical value is 3192 ms⁻¹).

To determine the I-term in practice, there are two variables to beestimated: (a) the angular frequency ω_(s) of stick-slip oscillations,and (b) the effective inertia J of the top drive. The latter isrelatively straightforward to determine and can either be calculatedfrom theoretical values of the gear inertia, the gear ratio and themotor rotor inertia, or it can be found experimentally by running anacceleration test when the top drive 30 is disconnected from the string.A typical formula for calculating top drive inertia J_(d) is:

J _(d) =J _(g) +n _(m) n _(g) ² J _(m)

where J_(g) is top drive inertia with the motor de-coupled (typicalvalue 100 kgm²), n_(g) is the gear ratio (>1), n_(m) of active motors(default value is 1), and J_(m) is the rotor inertia of the motor(typical value is 25 kgm²).

There are several ways that the angular frequency ω_(s) may beestimated, including: (i) calculations from string geometry, (ii) bymanual measurement (e.g. using a stop watch) and (iii) by automaticdetermination in the PLC software. An important advantage of the PItuning aspect of embodiments of the invention is that the damping effectof stick-slip oscillations is still obtained even if the estimate of thestick-slip period used to tune the PI controller is not very accurate.For example, FIG. 3 shows maximum damping occurring at a frequency of0.2 Hz. Even if the real stick-slip frequency is lower or higher thanthis, there is still a good damping effect (r˜0.8) obtained betweenabout 0.09 Hz and 0.4 Hz. Accordingly, the methods used to estimatestick-slip period do not have to be particularly accurate.

(i) String Geometry

It is possible to take a theoretical approach to determine thestick-slip period using parameters of the drill-string available on-sitein the tally book. A tally book is compiled on site for each drillstring and comprises a detailed record of the properties of each sectionof drill string (e.g. OD, ID, type of pipe), a section being defined asa length (e.g. 300 m) of the same type of drill pipe.

In the following it is assumed that the drill string 12 consists of onedrill pipe section of length l with a lumped bit impedance at the lowerend, represented by Z_(b). This impedance can be a pure reactive inertiaimpedance (iωJ_(b), where J_(b) is the inertia of the bottom holeassembly) or it can be a real constant representing the lumped damping(positive or negative) at the drill bit 28. The torque equations at thetop and at the bit represent the two boundary conditions. It can beshown that these two boundary conditions can be written as the followingmatrix equation.

$\begin{matrix}{{\begin{bmatrix}{\zeta + Z_{d}} & {\zeta - Z_{d}} \\{\left( {\zeta - Z_{b}} \right)^{{- }\; {kl}}} & {\left( {\zeta + Z_{b}} \right)^{\; {kl}}}\end{bmatrix} \cdot \begin{bmatrix}\Omega^{+} \\\Omega^{-}\end{bmatrix}} = \begin{bmatrix}0 \\0\end{bmatrix}} & (11)\end{matrix}$

where k is the wavenumber and Z_(d) is the impedance of the drillingmechanism.

No-trivial solutions to this system of equations exist if thedeterminant of the system matrix vanishes, that is, when

$\begin{matrix}{{^{\; 2\; {kl}} = {\frac{\left( {\zeta - Z_{d}} \right)\left( {\zeta - Z_{b}} \right)}{\left( {\zeta + Z_{d}} \right)\left( {\zeta + Z_{b}} \right)} = {r_{d}r_{b}}}}\;} & (12)\end{matrix}$

Here reflection coefficients at the drive r_(d) and at the bottom of thedrill string r_(b) have been introduced as follows:

$r_{d} = \frac{\zeta - Z_{d}}{\zeta + Z_{d}}$$r_{b} = \frac{\zeta - Z_{b}}{\zeta + Z_{b}}$

Notice that the top drive reflection coefficient r_(d)≈−1 for a stiffspeed controller (|Z_(d)|>>ζ) and the bit reflection coefficient r_(b)equals unity for a free lower end (Z_(b)=0).

The roots of equation (12) can be written as:

i2kl=ln(r _(d) r _(b))=ln|r _(d) r _(b) |+i(n2π+α_(d)+α_(b))  (13)

where m is a non-negative integer and α_(d) and α_(b) are the arguments(phase angles) of the complex reflection coefficients r_(d) and r_(b),respectively. The corresponding angular resonance frequencies are

$\begin{matrix}{\omega_{n} = {\left( {\alpha_{d} + \alpha_{b} + {n\; 2\; \pi} - {\; \ln {{r_{d}r_{b}}}}} \right)\frac{c}{2\; l}}} & (14)\end{matrix}$

Since, in general, the magnitudes and phases of the reflectioncoefficient are frequency dependent, the above equation is transcendent,without explicit analytic solutions. However, it can be solvednumerically by a PC or other computer.

The imaginary term of the above equation represents the damping of theeigenmodes. If |r_(d)r_(b)<1 the imaginary part of the root is positive,thus representing a normal, positive damping causing the time factorexp(iω_(n)t) to decay with time. In contrast, if |r_(d)r_(b)|>1 thedamping becomes negative, causing a small amplitude to growexponentially with time.

As an example, consider a case with a completely stiff speed controller(|r_(d)|=−1 and α_(d)π) rotating a drill string having a finite bottomhole inertia (Z_(b)=iωJ_(b), |r_(b)=1 and α_(b)=−2 tan⁻¹(ωJ_(b)/ζ)).Then the lowest (theoretical stick-slip) frequency ω_(s) becomes:

$\begin{matrix}{\omega_{s} = {\left( {\pi - {2\; {\tan^{- 1}\left( \frac{\omega_{s}J_{b}}{\zeta} \right)}}} \right)\frac{c}{2\; l}}} & (15)\end{matrix}$

With no extra bottom hole assembly inertia this expression reduces toω_(s)=πc/(2l). Notice that the resonance frequency decreases as theinertia J_(b) increases. In the extreme case when ω_(s)J_(b)>>ζ theabove formula can be rewritten as ω_(s)≈1√{square root over (J_(b)C)}where C=l/(GI_(p)) is the static compliance of the string. This is thewell-known formula for the natural frequency of a lumped inertia andspring system.

We have found that it is useful to study the relation between lower endspeed amplitude Ω_(s)≡Ω(x=l) and the corresponding top torqueT_(s)≡T(x=0). It can be shown from the equations above that this ratiois

$\begin{matrix}{\frac{\Omega_{s}}{T_{s}} = {\frac{{r_{d}{\exp \left( {{- }\; {kl}} \right)}} + {\exp \left( {\; {kl}} \right)}}{\zeta \left( {r_{d} - 1} \right)} = {{{- }\frac{\sin \; ({kl})}{\zeta}} - \frac{\left( {1 + r_{d}} \right)\cos \; ({kl})}{\left( {1 - r_{d}} \right)\zeta}}}} & (16)\end{matrix}$

Using the fact that characteristic impedance can be written as ζ≡kl/(ωC)the down hole speed amplitude can be expressed by

$\begin{matrix}{\Omega_{s} = {{{- \frac{\sin \; ({kl})}{kl}}{C \cdot {\omega}}\; T_{s}} - {\frac{\left( {1 + r_{d}} \right)\cos \; ({kl})}{\left( {1 - r_{d}} \right){kl}}C\; \omega \; T_{s}}}} & (17)\end{matrix}$

Notice the that the second term vanishes if the speed controller is verystiff (r≈−1) or when kl≈π/2. However if a soft speed controller is usedand there is a high inertia near the bit so that kl for the stick-slipfrequency is significantly less than π/2, then the second term may besignificant and should not be omitted.

The theory above can be generalized to strings with many sections andalso to cases with distributed damping. If a linear damping term isincluded, the generalization causes the wave number and characteristicimpedances to be complex and not purely real. If the string consists ofn uniform sections the general wave solution consists of 2n complexspeed amplitudes, representing pairs of up and down propagating waves.Continuity of angular speed and torsion across the section boundariescan be expressed by 2(n−1) internal boundary conditions, which add tothe two end conditions in equation (11). These can be set up as ahomogeneous 2n×2n matrix equation. The roots of this system of equationsare those frequencies making the system matrix singular. Although it ispossible to find an analytic expression for the system determinant, thesolutions are found numerically by a PC or other computer on site.IADC/SPE 15564 provides an example of one way to do this, and itscontent is hereby incorporated by reference for all purposes.

FIG. 4A′ and 4A″ show a typical window 50 available on the driller'sconsole that enables the driller to trigger a PC to estimate a newstick-slip period based on string geometry. In particular a table 52represents the sections of the drill string including BHA, heavy-weightdrill pipe (HWDP), and drill pipe sections 1 to 6. Available fields foreach section are: length, outer diameter and inner diameter. The drillerfirstly determines from the on-site tally book how many sections thedrill string is divided into. In this example the drill string has eightsections. For each section the driller enters figures into the threefields. A button 54 enables the driller to trigger a new stick-slipperiod to be estimated based on the string geometry entered in the table52. In particular, the table establishes the 2n×2n matrix equationmentioned above and the PL (not shown) uses a numeric method to find theroots of the matrix that make the matrix singular. The smallest root isthe stick-slip period output 56 in the window 50.

(ii) Manual Estimation

To determine the stick-slip period manually, the driller may observe thedrive torque as displayed on the driller's console 34 and determine theperiod by measuring the period of the variation of the drive torque witha stopwatch. This is readily done since each period is typically 2 s to10 s. An alternative method is for the driller to listen to the changein pitch of the top drive motor and to time the period that way. Asmentioned above, such methods should be sufficient as the estimatedstick-slip frequency does not have to be particularly close to the realstick-slip frequency in order that the stick-slip oscillations aredamped.

(iii) Automatic Estimation

Automatic estimation means that the PLC software estimates thestick-slip period or frequency from measurements made during drilling.In particular, the top drive torque signal is filtered by a band-passfilter that passes frequencies in the range 0.1 Hz to 0.5 Hz (i.e. aperiod of between 2 s and 10 s), that is the filter favours thestick-slip component and suppresses all other frequency components. ThePLC then detects the period between every new zero up-crossing of thefiltered torque signal and uses these values in a recursive smoothingfilter to obtain a stable and accurate period estimate. The finalsmoothing filter is frozen when either the stick-slip severity (seebelow) falls below a low critical value, or the tuning method isactivated.

To help the period estimator to quickly find the accurate period, theoperator can either put in a realistic starting value or pick atheoretical value calculated for the actual string (determined as perString Geometry section above).

In use, the tuned PI controller is activated when there is a significantstick-slip motion (as determined by the driller or by software).However, the stick-slip frequency estimation (period measurement) takesplace before the tuned PI controller is actually used to control thedrilling mechanism. Once complete the period estimator is turned offwhen PI controller is on, because the natural period of the stick-sliposcillations can change slightly when soft speed control is used.

There does not appear to be a need for very frequent retuning of theestimated frequency because the natural stick-slip frequency variesslowly with drill string length. It is a good idea, however, toautomatically update the period at every connection i.e. when another 30m of drill pipes are added to the drill string. To do that it ispossible to use theoretical sensitivity analysis to predict how thestick-slip period increases with drill string length. One way to do this(but not the only way) is to find the theoretical periods for two stringlengths (L and L+200 m, say) and then use interpolation for the increasecaused by the addition of a 30 m section in order to update theestimated period.

Estimation of Stick-Slip Severity and Instantaneous Bit Speed

An additional aspect of some embodiments of the invention is provided asa set of computer executable instructions in the PLC software thatenables quantification of bit speed variations and an estimate of theinstantaneous bit rotation speed. ‘Bit speed’ means the BHA rotationspeed excluding the contribution from an optional mud motor. This aspectmay be provided separately from or in combination with the PI controllertuning.

This estimation is achieved by combining the known torsional complianceC of the drill string and the variations of the drive torque. Ingeneral, since the torque is not a strictly periodic signal but oftenpossesses a wide range frequencies, an accurate calculation is extremelycomplicated and is therefore not suitable for implementation in a PLC.However, we have realised that since the stick-slip motion is dominatedby the fundamental stick-slip frequency, it is possible to achievefairly good estimates based on this frequency only.

The key equation is (17) above, which describes a good approximation forthe complex speed amplitude as a function of the top string torque. Thetwo terms in this expression must be treated differently because theyrepresent harmonic components having a 90 degrees phase difference.While the imaginary factor iωT_(s) should be treated as the timederivative of the band pass filtered torque, the real term factor ωT_(s)can be approximated as the product of the band pass filtered torque andthe stick-slip frequency. Since the band pass filter suppresses allfrequencies except the stick slip-frequency, it is possible tosubstitute direct time integration by an integration basedapproximation. This approximation is based on the fact that iω≈−ω_(s)²/(iω), where 1/(iω) represents time integration. This approximationfavours the stick-slip frequency and suppresses higher harmonics. Thetime domain version of (17) suitable for implementation in the PLC 38is:

$\begin{matrix}\begin{matrix}{\Omega_{b} = {{{- \frac{\sin \; ({kl})}{kl}}{C \cdot \frac{T_{bp}}{t}}} - {\frac{\left( {1 + r_{d}} \right){\cos ({kl})}}{\left( {1 - r_{d}} \right){kl}}C\; \omega_{s}T_{bp}}}} \\{\approx {\frac{\sin \; ({kl})}{kl}{C \cdot \omega_{s}^{2}}{\int{T_{bp}{t}}}}}\end{matrix} & (18)\end{matrix}$

Here the phase parameter kl=ω_(s)l/c. In the last approximation theintegral approximation for time derivation is used and the second termis omitted.

Even though the formula above is based on a single section string,simulations have shown that it also provides good estimates formulti-section strings if the total string compliance C is used:

$\begin{matrix}{C = {\sum\limits_{j = 1}^{m}\frac{l_{j}}{I_{p,j}G}}} & (19)\end{matrix}$

A version of the algorithm implemented in the PLC 38 to estimate bothinstantaneous BHA speed and a stick-slip severity, comprises thefollowing steps.

1. Estimate the string torque by correcting for inertia effects(subtract the effective motor inertia times the angular acceleration)and by using the gear ratio to scale it properly;

2. Band pass filter the estimated torque with a band pass filter centredat the observed/estimated stick-slip frequency. The filter should be of2nd order or higher, but can preferably be implemented in the PLC as aseries of 1st order recursive IIR filters;

3. Calculate the total static drill string compliance using equation(19) above;

4. Calculate the phase parameter kl=ωl/c where ω_(s) is the determinedangular stick-slip frequency;

5. Calculate the dynamic downhole speed by using either the accurate orthe approximate version of equation (18) above;

6. Calculate the “stick-slip severity” σ, which is the normalizedstick-slip amplitude, determined as the ratio of dynamic downhole speedamplitude over the mean top drive rotational speed;

7. Find the instant speed as the sum of the low pass filtered top drivespeed and the estimated dynamic downhole speed. Clip to zero if theestimated speed goes negative;

8. Output data to be plotted on a graph (e.g. RPM versus time) shown ona display on a driller's console for example;

9. Repeat steps 1 to 8 to provide substantially real-time estimate ofbit speed.

It is envisaged that this method could be performed where only the BHAspeed estimate is output or only the stick-slip severity is output.

Regarding step 6, a possible way of estimating the stick-slip severityis to use the following formula where LP( ) denotes low pass filtering:

$\begin{matrix}{\sigma = \frac{\sqrt{2 \cdot {{LP}\left( \Omega_{b}^{2} \right)}}}{\Omega_{set}}} & (20)\end{matrix}$

Because the above method takes the reflection coefficient into account,it applies both for a standard and tuned speed control. Duringacceleration transients when the top drive speed is changedsignificantly the estimator is not reliable but can give large errors.Nonetheless we believe this is a useful tool for assessing downholeconditions, either automatically in software or by display for analysisby a driller.

The ratio of dynamic speed amplitude to the average top drive speed is adirect and quantitative measurement of the stick-slip motion, moresuitable than either the dynamic torque or the relative torqueamplitude. Even though the estimated bit speed is not highly accurate,it provides a valuable input to the driller monitoring of it in a trendplot will give the operator more explicit information on what ishappening at the bit.

User Interface

A user interface is provided for the driller's console 34 that comprisesa graphical interface (see FIGS. 4A′ and 4A″, and 4B′ and 4B″) whichprovides the operator with direct information on the stick-slip status.Stick-slip is indicated by three different indicators:

A “traffic light” indicator 58 in FIG. 4A′ with 3 levels of stick-slip:a green light for small amplitudes (0-30%), a yellow warning light ifthe speed oscillations are significant (30-70%) and finally a red lightif even higher amplitudes are estimated. This percentage value is basedon the stick-slip severity as determined above.

The stick-slip severity is plotted in a plot 62 of torque versus time inFIG. 4B′ and 4B″ to see how the stick-slip has developed over aspecified period of time.

The instant bit speed estimate in a plot 64 of instantaneous bit speedversus time in FIG. 4B′ and 4B″ giving a visual and direct impression ofthe down hole stick-slip status.

As mentioned above, the window 50 requires the operator to input a roughdescription of the string, in terms of a simplified tally. This tallyaccepts up to 8 different sections where the length, outer diameter andmass per unit length are specified. This information is used forcalculating both the theoretical estimated frequency for the lowest modeand the static drill string compliance at this frequency.

The operator can switch the tuned PI controller on or off. In the offstate, the standard drive speed controller is used. When the tuning isturned on, this speed controller is bypassed by the tuned PI controller42 which is implemented in the PLC 38. If the drive controller in thetop drive 30 is a modern digital one, it is also possible to changedrive speed controller itself, instead of bypassing it. However, if thebypass method is chosen, this is achieved by sending a high speedcommand from the PLC 38 to the speed controller in the top drive 30 andby controlling the output torque limit dynamically. In normal drillingthis torque limit is used as a safety limit preventing damage to thestring if the string suddenly sticks. In the tuned control mode, whenthe PLC 38 controls the torque limit dynamically, this limit issubstituted by a corresponding software limit in the PLC 38.

The operator can also change the prevention or mobility factor a withinpreset limits via buttons 60, typically between 0.05 and 0.33. A highfactor implies a softer speed control and less probability for thestick-slip motion to start or persist. The disadvantage of a high factoris larger fluctuations of the top drive speed in response to harmlesschanges in the string torque level. It may be necessary to choose a highfactor to cure severe stick-slip oscillations but the operator shouldreduce the factor when smooth drilling is restored.

It is envisaged that the decision to activate and de-activate the tunedspeed control may be taken by the PLC 38 or other electronic controller.Such a controller may monitor the instantaneous estimate of bit speed asset out above. When a period pattern of stick-slip is observed, thecontroller may activate the tuning. Furthermore the controller maygradually increase the mobility or prevention factor to increase thesoftness of the top drive 30 if the stick-slip oscillations do notreduce in magnitude over a predetermined period e.g. 2 minutes. Once thestick-slip oscillations have reduced or substantially disappeared thecontroller may gradually reduce the mobility factor (e.g. down to a=0.1)to improve drilling efficiency.

HIL Testing

The PI tuning method has recently been extensively tested in so-calledHardware In the Loop (HIL) simulations. In these tests the PLC programsare run on a physical PLC interfacing to a real-time simulation model ofthe drive and the drill string.

The simulation model being used for the HIL testing of tuning method hasthe following features:

1. The drive is modelled as a standard PI speed controller with torqueand power limitations and anti-windup. The torque or current controlleris perfect in the sense that the actual torque is assumed to match theset torque with no delay.2. The model can handle a plurality of drive motors connected to theoutput shaft by a gear.3. The drill string is modelled as a series of lumped inertia and springelements derived from any tally book. The grid length used in mostexamples below is approximately 28 m, which is the typical length of atriple stand. Hence the 3200 m long string used below consists of 114elements.4. The static friction torque is calculated for every element, based onthe theoretical contact force being a function of weight andinclination, curvature and tension. The effect of WOB and bit torque isalso included.5. The dynamic, speed dependent friction torque is modelled as a sum ofthree terms. The first term is a soft-sign variant of the Columnfriction, the second represents and extra static start friction and thethird is a linear damping term, independent of the contact force. Tosimulate instability with growing oscillation amplitude from smoothdrilling, this damping coefficient must be negative.

The model was first developed as a Simulink model under the Matlabenvironment. It is later implemented with the Simulation Module toolboxunder the National Instrument LabView environment and run on a powerfulPC platform. Although this PC is not using a real time (RT) operativesystem, its high power makes the model RT for all practical purposes.

The LabView simulation program is linked to the PLC by a so-calledSimbaPro PCI profibus DP (Distributed Peripherals) card, which cansimulate all DP nodes connected to the PLC. The update time is set to 10ms (100 Hz), which is within the PLC cycle time (typically 20 ms).

Results from the HIL testing are shown in FIG. 5. The string used is3200 m in length similar to the string used in the field test (seebelow). The theoretical period for the lowest mode is 5.2 s. FIG. 5shows a graph 70 of the torque and speed for the drill string (trace 72)and for the top drive (trace 74) during a 150 s period including a 5 sinterval where the top drive speed is accelerated from zero to 100 rpm.The tuned speed control is turned on 30 s after start of rotation.Steady stick-slip oscillations are established soon after the start up.The stick-slip period stabilizes around 5.3 s. This is slightly longerthan the theoretical pendulum period, but the extended period isconsistent with the fact that the sticking interval is substantial. Notethat the top drive speed is nearly constant during this part of thespeed control.

When the tuned speed control is turned on, the top drive speed (trace78) temporarily shows a pronounced dynamic variation 79 in response tothe large torque variations. But after a few periods the stick-slipmotion fades away and the top drive speed, as well as the bit speed,become smooth. When tuned speed control is turned off again, thedown-hole speed (trace 76) amplitude starts to grow, until fullstick-slip motion is developed. This instability is a consequence of thenegative damping included in the string torque model.

FIG. 6 shows results 80 from the same simulations, but now with focus onthe PLC estimated stick-slip severity (trace 87) and instantaneous bitspeed (trace 84)—note that the lower graph is a continuation of theupper graph and shows the difference between simulated speed (trace 84)and estimated speed (trace 86). The bit speed estimate is fairly goodduring steady conditions but has significant error during start-up.Despite this, the estimated bit speed is able to provide the drillerwith a useful picture of down hole speed variations. The effectivenessof the tuned speed controller is clearly illustrated by the trace 87 ofstick-slip severity: when tuned speed control is in use, the stick-slipseverity falls almost to zero. Once tuned control is switched off, thestick-slip severity increases once again.

Field Test

The tuning has been tested in the field, while drilling a long deviatedwell. The string was approximately 3200 m long with a 5.5 inch drillpipe. Unfortunately, the test ended after a relative short period ofsevere stick-slip conditions, when the PDC bit drilled into a softerformation. The new formation made the bit less aggressive with lessnegative damping, thus removing the main source of the stick-sliposcillations.

FIG. 7 shows an example where stick-slip motion is developed whilerotating with the standard stiff speed controller. Two graphs 90 areshown: one of drive torque versus time, and the other of bit speedversus time. A few comments on these graphs are given below:

The data was recorded from the PLC at a sampling rate of approximately 9Hz.

The “TD corrected” torque (trace 92) is the estimated string torque andequal to the measured drive torque corrected for inertia effects.

The TD corrected torque as well as the bit speed are estimated by postprocessing the recorded data using the methods described above.

The standard top drive speed controller is very stiff, becausevariations of the measured speed (trace 94) can barely be seen afterturning off the tuned speed control and the top drive rpm is virtuallyconstant. The corresponding small accelerations are the reason why themeasured drive torque almost matches the inertia corrected string torqueduring this period.

The high frequency torque oscillations (at 1.1 Hz) seen during firstpart of the trace 96 when tuning is on probably come from a higher moderesonance in the drill string. These vibrations seem to be independentof the type of speed controller used, but they vanish when stick-slip isdeveloped.

The prevention factor (line 98) is the operator set mobility factor amentioned above.

The observed stick-slip period is approximately 5.2 s, which is in goodagreement with the theoretical period for this particular string.

Another example of successful curing of stick-slip motion is shown inFIG. 8. In this figure a similar graph 100 to graph 90 is shown:

The “TD set” speed (trace 102) is the speed command sent to the drive.When the tuning is turned on, this level is raised so the bypassed drivespeed controller always tries to increase the torque beyond the dynamiclimit of the new speed controller. In this case the speed increase is aslightly too small, causing the dynamic speed to be clipped by the drivespeed controller. This clipping will reduce the damping effect under thetuned PI controller.

When tuning is turned on, the mobility factor (line 104) isapproximately 15%. This is a little too low, because stick-sliposcillations are not cured before the operator increases this factor at106.

After the stick-slip motion has faded at about 4310 s, the 1.1 Hzoscillations reappear with an amplitude similar to what was observedbefore. But now the vibrations are seen also in the measured speed.

Additional data, not included here, show that the 1.1 Hz oscillationamplitudes decrease but do no vanish completely when the mobility factoris further increased. It means that even though the top drive impedanceis inertia dominated at this frequency the soft PI controller also hassome dampening effect on higher mode oscillations as well.

Higher Stick-Slip Modes

The stick-slip damping method described above works very well in a widerange of cases. However, extensive Hardware-In-the-Loop (HIL) simulationtesting has revealed a further problem when the string is extremely long(typically 5000 m or more) and the measured i.e. fundamental stick-slipperiod exceeds about 5-8 s i.e. a frequency ω_(s) of about 0.2-0.13 Hz.The method is still able to damp the fundamental mode stick-sliposcillations, but as soon as these oscillations are dampened, the secondnatural mode tends to become unstable and grow until full stick-slip isdeveloped at that second mode. This second mode has a natural frequencywhich is approximately three times higher than the fundamentalstick-slip frequency ω_(g). The higher order stick-slip oscillations areseen as short period (less than about ⅓ of the fundamental stick-slipperiod) and large amplitude (greater than about 2 kNm) cyclic variationsof the drive torque. We have also found via simulations that, duringsecond mode stick-slip oscillations, the bit rotation speed variesbetween zero and peak speeds exceeding twice the mean speed.

Simulations have shown that the system sold under the trade mark SOFTTORQUE® also suffers from the same problem. Neither system is able todamp at the same time both the first and second mode stick-sliposcillations.

We have discovered that by reducing the effective inertia of thedrilling mechanism this problem can be addressed. There are several waysthat the effective inertia can be reduced including making a relativelysmall change in the tuned PI controller described above, and selecting ahigher gear in the drilling mechanism. Advantages of reducing theeffective inertia include: more effective damping of higher modes, andincreased tolerance in the method to errors in the estimated stick slipfrequency. There are two ways we have identified to reduce the effectiveinertia of the drilling mechanism: by tuning of the speed controller andby changing gear (if possible). Each will be described below.

Speed Controller Tuning to Dampen Higher Modes

For clarity, the first embodiment of the speed controller 42 describedabove will be referred to as the ‘tuned PI controller’ and the secondembodiment of the speed controller 42 described below will be referredto as the ‘inertia compensated PI controller’.

Before describing the optional improvement to the method, we first setout a basic description of the higher modes of torsional stringoscillations. As described above (see equations (14) and (15)) thenatural angular frequency for mode m of a lossless string rotated by topdrive with zero mobility is given by:

$\begin{matrix}{\omega_{m} = {\left( {{2\; m} - 1 - {\frac{2}{\pi}{\tan^{- 1}\left( \frac{\omega_{m}J_{b}}{\zeta} \right)}}} \right)\frac{\pi \; c}{2\; l}}} & (21)\end{matrix}$

where

m is a positive integer indicating mode number (m=1 for the lowestmode);

J_(b) is the inertia of the bottom hole assembly (BHA);

ζ is the characteristic impedance of the drill pipes;

c is the sonic speed for torsional waves; and

l is the length of the drill pipe section (excluding the BHA length).

It is convenient to introduce the normalized frequency

$\begin{matrix}{\varphi_{m} = {\frac{2\; l}{\pi \; c}\omega_{m}}} & (22)\end{matrix}$

and the normalized inertia

$\begin{matrix}{j_{b} = {{\frac{\pi \; c}{l\; \zeta}J_{b}} = \frac{J_{b}}{J_{dp}}}} & (23)\end{matrix}$

Here we have used the fact that the characteristic impedance can beexpressed as ζ=J_(dp)c/l, where J_(dp) represents the total inertia ofthe drill pipes. The frequency equation (21) can now be written as:

$\begin{matrix}{\varphi_{m} = {{2m} - 1 - {\frac{2}{\pi}{\tan^{- 1}\left( \frac{\varphi_{m}j_{b}}{2} \right)}}}} & (24)\end{matrix}$

For non-zero inertia values this equation is transcendental, which meansthat it has no explicit analytical solutions and must be solvednumerically, as described above. Referring to FIG. 9 a graph 110 showsthe three lowest roots (m=1, 112; m=2, 114; m=3, 116) versus thenormalized inertia j_(b). Frequency ratio curves 118, 119 show that theratio is nearly constant and approximately equal to 2m−1 for small BHAinertia (j_(b)≦1). In practice, very long drill strings (>5 km) used forextended horizontal reach, have quite small and light BHAs (withoutdrill collars or heavy weight drill pipes) to limit the total frictiontorque. Therefore, the low inertia ratios 3 and 5 for the second andthird modes respectively are very good approximations to reality.

The corresponding mode shapes for the dynamic rotation speed can befound from the wave numbers k_(m), which can be written as:

$\begin{matrix}{k_{m} = {\frac{\omega_{m}}{c} = {\frac{\pi}{2\; l}\varphi_{m}}}} & (25)\end{matrix}$

The corresponding eigenfunctions describing how the angular speedamplitude varies with depth, are

$\begin{matrix}{\sigma_{m} = {{\sin \left( {k_{m}z} \right)} = {\sin \left( \frac{{\pi\varphi}_{m}z}{2\; l} \right)}}} & (26)\end{matrix}$

where z is depth referred to the top drive position.

Referring to FIG. 10 a graph 120 shows the mode shapes m±0.256σ_(m) forthe three lowest modes for the case when j_(b)=1. The Y-axis representsnormalized depth z_(n)=−z/l. It is apparent that the bit (lower stringend where z=−1) is close to an anti-node, for all three modes.

We have discovered that the speed controller 42 can be improved tocounter stick-slip of the drill bit at both the first and second modesand, to some extent, higher modes of stick-slip oscillation. The basisfor the improvement is found in the equation of angular motion of thedrilling mechanism 30. Referring to equation (3) above, the equation ofmotion for the drilling mechanism 30 can be expressed by:

${J_{d}\frac{\Omega}{t}} = {T_{d} - T}$

where the torque T_(d) of the drilling mechanism is given by:

$\begin{matrix}{T_{d} = {{J_{c}\frac{\Omega}{t}} + {P\left( {\Omega_{set} - \Omega} \right)} + {I{\int_{\;}^{\;}{\left( {\Omega_{set} - \Omega} \right)\ {t}}}}}} & (28)\end{matrix}$

and where

J_(d) is the total mechanical drive inertia (including gear and drivemotors);

J_(c) is a computer or manually controllable compensation inertia;

P is the speed controller P-factor (referred to output shaft);

I is the speed controller I-factor;

Ω_(set) is the angular set speed; and

Ω is the actual top drive speed as measured.

Thus the speed controller uses three terms to control the torque T_(d)applied by the drilling mechanism 30 to the drill string. The second twoterms on the right-hand side are familiar from equation (2) above.

The first term on the right-hand side of equation (28) is the keycomponent for extending the functionality of the tuned PI controller ofthe first embodiment. In contrast to a normal derivative term of a PIDcontroller, which is proportional to derivative of the speed error, thenew speed controller term is proportional to the derivative of themeasured speed only. The proportionality factor J_(c) is called thecompensation inertia because it has dimensions of inertia and it reducesthe effective inertia of the drilling mechanism 30. This is seen bycombining equations (2) and (28), and moving this derivative term overto the left hand side:

$\begin{matrix}{{\left( {J_{d} - J_{c}} \right)\frac{\Omega}{t}} = {{P\left( {\Omega_{set} - \Omega} \right)} + {I{\int_{\;}^{\;}{\left( {\Omega_{set} - \Omega} \right)\ {t}}}} - T}} & (29)\end{matrix}$

This is the equation of motion for a drilling mechanism 30 with areduced inertia using a conventional or tuned PI speed controller. Theadvantage of this inertia reduction is that the absorption bandwidth ofthe drilling mechanism 30 is increased, as explained below. Furthermore,since J_(c) is software controllable, inertia compensation can beswitched on and off readily in the speed controller and, when on, can beadjusted either in real-time if needed. Alternatively, it is possible toallow the driller to set J_(c) manually via the driller's console forexample.

Following the same methodology as described above in conjunction withequation (6) above, the effective torsional impedance can be written asthe complex function:

$\begin{matrix}{Z_{d} = {{{\omega}\left( {J_{d} - J_{c}} \right)} + P + \frac{I}{\omega}}} & (30)\end{matrix}$

where i=√{square root over (−1)} is the imaginary unit and ω is theangular frequency. The corresponding reflection coefficient r_(d) forthe drilling mechanism 30 is

$\begin{matrix}{r_{d} = {- \frac{P - \zeta + { \cdot \left( {{\omega \left( {J_{d} - J_{c}} \right)} - \frac{i}{\omega}} \right)}}{P + \zeta + { \cdot \left( {{\omega \left( {J_{d} - J_{c}} \right)} - \frac{I}{\omega}} \right)}}}} & (31)\end{matrix}$

where ζ is the so-called characteristic impedance of the drill pipe andrepresents the ratio of torque and angular speed for a progressive wavepropagating along the drill string 12. This complex reflectioncoefficient represents both amplitude and phase of the reflected wavewhen a unit incident torsion wave, which propagates upwards in the drillstring 12, is reflected at the top. The magnitude of this reflectioncoefficient is strongly related to the torsional oscillations asdescribed above in conjunction with the tuning of the speed controller42 to dampen the fundamental stick-slip oscillation.

It is convenient to define the effective inertia as J=J_(d)−J_(c) and anon-dimensional reactance b=(ωJ−I/ω)/P. The mobility parameter a=ζ/P isas defined above in connection with the first embodiment. The damping,which is the amount of torsional energy absorbed by the drillingmechanism 30 (i.e. the torsional energy not reflected back down thedrill string 12), then can be written as

$\begin{matrix}{{1 - {r_{d}}} = {1 - \sqrt{\frac{\left( {1 - a} \right)^{2} + b^{2}}{\left( {1 + a} \right)^{2} + b^{2}}}}} & (32)\end{matrix}$

When b=0, that is when ω=ω₀ ≡√{square root over (I/J)}, then the dampingis at its maximum and equal to 1−|r_(d)|=2a/(1+a). It can be shown thatthe damping equals half this value when b²=(1+a)²(2−a)/(2+a) and whenthe angular frequency is

$\begin{matrix}{\omega = {\sqrt{\omega_{0}^{2} + \left( \frac{b\; \zeta}{2{aJ}} \right)^{2}} \pm \frac{b\; \zeta}{2{aJ}}}} & (33)\end{matrix}$

The frequency ratio ω/ω₀ for the highest root (+ sign) gives aquantitative measure for the absorption bandwidth β of the drillingmechanism 30:

$\begin{matrix}{\beta = {\sqrt{1 + \left( \frac{b\; \zeta}{2a\; \omega_{0}J} \right)^{2}} + \frac{b\; \zeta}{2a\; \omega_{0}J}}} & (34)\end{matrix}$

This formula shows that the absorption bandwidth β is increased when theeffective inertia J is reduced. Accordingly following equation (9)above, the I term of the inertia compensated PI controller is set asI=ω_(s) ²J where J=J_(d)−J_(c) i.e. is set as an inertia compensatedvalue. When the I-term of speed controller 42 is set in this way, itcauses the drilling mechanism 30 to have an increased absorptionbandwidth on torsional vibrations compared to the tuned PI controller,since the latter is tuned primarily to dampen the fundamental stick-slipmode.

It is easily verified that the ratio between the highest and lowestroots of the frequency equation (33) equals β², meaning that thereflection curve is symmetric about the centre frequency when plottedwith a logarithmic frequency axis.

Referring to FIG. 11 a graph 130 illustrates the increase of absorptionbandwidth and shows the reflection coefficient versus frequency for astandard stiff speed controller 132, a tuned PI controller 134, and aninertia compensated PI controller 136. The example assumes a 5″ drillpipe (having a characteristic drill pipe impedance of ζ=340 Nms), atotal mechanical top drive of inertia J_(d)=2000 kgm² (i.e. the sum ofthe mechanical inertia due to the motor and gear), a 50% inertiacompensation (J_(c)=J=0.5J_(d)), a mobility parameter of a=0.25, and anobserved or measured fundamental (m=1) stick-slip frequency of 0.1 Hz(period=10 s). By reducing the effective inertia of the drillingmechanism 30, the absorption bandwidth is increased from β=1.76 (tunedPI controller) to β=2.75 (inertia compensated PI controller). The arrowsin FIG. 11 are positioned at the natural frequencies of the respectivemodes. It is clearly seen that the reflection coefficient for the secondmode (m=2) drops from 0.93 to 0.82 when switching from the tuned PIcontroller to the inertia compensated PI controller. This droprepresents a large damping improvement, sufficient to inhibit, and incertain embodiments prevent, second mode stick-slip oscillations.

It is also possible, optionally, to further improve the damping ofhigher modes (i.e. m≧2) by shifting the minimum of reflectioncoefficient curve to higher frequencies. This is a kind of controlledde-tuning in which the maximum damping frequency is deliberately movedaway from fundamental frequency of stick-slip oscillations as measuredor observed (see section above on measurement of ω_(s)). Once thefundamental frequency has been measured or observed, the value can beincreased by between about 0% and 40%. This shifted fundamentalfrequency is then used to determine the I-term of the speed controlleras described above. The effect of this is that the reflectioncoefficient curve is shifted to higher frequencies, thereby reducing thereflection coefficients of at least some of the higher modes ofstick-slip oscillation. An alternative way to determine the increase inω_(s) is by some power of β between zero and one, β^(1/4) for example. Aparticular advantage of this is that the damping of the fundamental moderemains near to its original value, for example a change in thereflection coefficient from 0.6 to 0.62.

However, care has to be taken to ensure that the minimum of thereflection coefficient curve is not shifted too far from the fundamentalmode of the stick-slip oscillations. We suggest that the fundamentalstick-slip frequency used to determine the I-term is not increased bymore than a factor β^(1/2) above the actual measured or observedfrequency. In this way damping of at least some of the higher modes(e.g. m=2,3) is improved whilst sacrificing only a small amount ofdamping of the fundamental stick-slip mode.

A further advantage of shifting the minimum reflection point (i.e.maximum damping) to higher frequencies is that the damping offrequencies below the fundamental is increased. This means thatvariations in bit torque cause smaller variations in angular speed atthe top of the drill string 12 making the drilling mechanism appear“stiffer” at these low frequencies, which is important for drillingefficiency.

Referring to FIG. 12 a graph 140 illustrates an example of suchcontrolled de-tuning. The reflection curve 142 of an inertia compensatedspeed controller has been de-tuned so that the maximum damping frequencyis about 22% higher than the fundamental stick-slip frequency (shown bythe reflection curve 144 of a speed controller tuned primarily to dampenthe fundamental frequency). In the reflection curve 142 the reflectioncoefficient at the fundamental frequency has increased slightly, from0.6 to 0.62, while the second mode reflection coefficient has beensignificantly improved from 0.82 to 0.75.

Somewhat surprisingly we have found that using de-tuning only i.e.shifting the fundamental damping frequency but keeping inertiacompensation constant, lead to a narrower absorption bandwidth with avery small shift of the high frequency part. In order to overcome thiswe found that combining a frequency shift with extra inertiacompensation achieved both at the same time: the frequency was shiftedwhilst preserving the wider absorption bandwidth so that damping of oneor more higher mode was improved. One way to do this is to keep theproduct ω₀J constant. In the example shown in FIG. 12 the effectiveinertia is divided by the same factor, β^(1/4)=1.22, by which the centrefrequency is increased. This choice leaves the product of ω₀J and thebandwidth parameter β unchanged. This kind of frequency shift impliesthat the inertia compensation is increased, in this particular case fromJ_(c)=0.5J_(d) to J_(c)=0.59J_(d).

The analysis above is based on the assumption that there is no timedelay or filtering of the measured speed Ω. In practice, a speedmeasurement will be associated with a small time delay. Furthermore, thedrive acceleration needed for inertia compensation can be very noisedriving unless the derivative filter is combined with a filter.Referring to FIG. 13 a graph 150 shows the effect of a 20 ms delay ofthe measured speed Ω and a low pass filter (time constant 50 ms) used toproduce a smoothed acceleration signal. From this figure it is seen thatthe delay and filter affects the reflection coefficient of the inertiacompensated controller so that it exceeds unity for high frequencies(>0.75 Hz). This means that frequencies have negative damping and willgrow in amplitude unless the natural damping along the string exceedsthe negative contribution from the drilling mechanism 30. In contrast,the delay effect on the reflection coefficients for the normal and tunedcontrollers is very small. So whilst the compensation inertia J_(c) isadjustable (e.g. higher J_(c) increased absorption bandwidth andvice-versa), care has to be taken when increasing it. In particular, asthe absorption bandwidth increases, a wider range of frequencies aresubjected to negative damping.

To implement the new torque term from equation (28), the PI controllerrequires angular acceleration as an input signal. The angular driveacceleration is normally not measured separately but derived from thespeed signal by using the following difference approximation

$\frac{\Omega}{t} \approx \frac{\Delta \; \Omega}{\Delta \; t}$

Here ΔΩ is the measured speed change during the computing cycle time.This approximation introduces a delay time (equal to half the cycletime), in addition to a possible delay in the measured speed itself.

Optionally, the speed controller 42 may be configured to check theapproximate fundamental stick-slip period as determined or measured,against a period threshold such as 5 s. If the fundamental period isgreater than this threshold, the speed controller may reduce theeffective inertia of the drilling mechanism 30 to dampen any higher modeoscillations. Furthermore the amount of damping may be proportional tothe fundamental period, for example starting a 0% for a fundamentalperiod of 5 s, increasing linearly to 75% inertia compensation for afundamental period of 8 s. Other adjustments (e.g. non-linear) ofeffective inertia with measured period are envisaged.

Referring to FIG. 14 a graph 160 illustrates how the inertia compensatedspeed controller 42 is able inhibit second mode stick slip oscillations.The upper subplot 162 shows top drive speed 163 and the bit speed 164when a tuned PI controller is activated 50 s after start of drill stringrotation. The stick-slip oscillations at the fundamental frequency arecured, but after a short transient period 165 second mode stick-sliposcillations 166 appear. Note that the second mode frequency is nearly0.3 Hz, or three times higher than the fundamental mode frequency.

The lower subplot 167 shows the results from a similar simulation whenan inertia compensated PI controller is activated after 50 s from thestart. The improved speed controller has used a compensation factor of0.5, that is J_(c)=0.5J_(d), but no frequency shift (or “de-tuning”) isapplied. This speed controller is able to prevent stick-sliposcillations at both the fundamental and second modes, resulting insmooth drilling with only small variations of the drive torque and thebit speed.

Changing Gear to Dampen Higher Modes

If the drilling mechanism 30 has a multiple speed gear box, the gearselection also affects the absorption bandwidth and the damping ofhigher modes in a similar way as the tuning method above. This isdeduced from discussion above and from the expression of the totalmechanical drilling mechanism inertia

J _(d) =J _(g) +n _(m) n _(g) ² J _(m)  (35)

whereJ_(g) is the gear inertia (referred to output shaft);J_(m) is motor (rotor) inertia;n_(m) is the number of motors; andn_(g) is the gear ratio (motor speed/output speed).

Switching from low gear to high gear implies that the gear ratio n_(g)drops, typically by a factor of two. As an example, consider a singlemotor top drive (n_(m)=1) with a motor rotor inertia of J_(m)=25 kgm², agear inertia of J_(g)=200 kgm² and with two gear ratios n_(g1)=8.49 andn_(g2)=4.25. The corresponding drive inertia values then becomesapproximately J_(d1)=2000 kgm² and J_(d2)=650 kgm² in low and highgears, respectively. The reduction in mechanical inertia represents apronounced increase of the absorption bandwidth as seen in FIG. 15,actually from β=1.76 (low gear) to β=3.95 (high gear). In FIG. 15 trace172 shows the reflection coefficient versus frequency for an untunedstiff controller in high gear, trace 174 for a tuned PI controlleraccording to the first embodiment in low gear, and trace 176 for thesame tuned PI controller in high gear. The increase in absorptionbandwidth at the higher gear can be seen clearly.

In practice, the possibility of selecting a high gear (i.e. with lowinertia) is limited, both because many top drives do not have multiplespeed gear boxes, and because the torque capacity in high gear may betoo low to overcome the high friction torque in extremely long anddeviated wells.

Using a PID-Type Speed Controller to Dampen Higher Modes

Another alternative is that the inertia compensation can be implementedthrough a digital PID-type speed controller of the type found inindustrial AC drives (e.g. the ACS800 manufactured by ABB). Such drivestypically have an interface which allows manual control of the P, I andD terms of the speed controller. The terms are set according to equation(28) and in particular, the P and I terms may be set as described above.However, the D term is more complicated to implement because it isproportional to the derivative of the speed of the drive, rather than tothe derivative of the speed error of the drive as in normal PID control.Therefore it is believed that it is not possible to implement the newterm

$J_{c}\frac{\Omega}{t}$

via the standard D-term because this latter term will have an unwantedeffect on the set speed. In particular, the D term will need to be setas a negative value in order to reduce the effective inertia. However, astandard digital PID controller can be adapted by adjustment of thespeed controller firmware via the low level source code of the drive or,if that is inaccessible to the user, by requesting the manufacturer ofthe drive to implement this term in the firmware.

It is to be noted that the three terms in a standard PID controller arenot always specified directly. Instead they are commonly specifiedindirectly through a so-called k-factor, which is a normalized P-factor,a time integration constant t_(i) and a derivative time constant t_(d).The P-factor (referred to the motor axis, has the dimensions of Nms andis related to the k-factor by P_(m)=k*T_(nom)/(π*N_(nom)/30) whereT_(nom) (in Nm) is the nominal motor torque and N_(nom) (in rpm) is thenominal motor speed (usually found on the name plate of the motor). Theintegration time constant is the ratio of t_(i)=P/I while the derivationtime constant is t_(d)=D/P.

In summary, there is described a PI or PID controller tuning method forinhibiting detrimental stick-slip oscillations. In a first embodiment, aspeed controller is provided that enables a drilling mechanism to absorbenergy from stick-slip oscillations over an absorption bandwidth thatincludes a fundamental frequency of those oscillations. In a secondembodiment, a speed controller is provided in which the absorptionbandwidth of the drilling mechanism is increased, and the energyabsorption of higher mode(s) is improved over the first embodimentsufficient to inhibit both the fundamental and one or more higher modeof oscillation.

In the first embodiment, the system comprises a PI type drive speedcontroller being tuned so that it effectively dampens torsionaloscillations at or near the stick-slip frequency. It is passive in thesense that it does not require measurement of string torque, drivetorque or currents, as alternative systems do. The dampingcharacteristics of a tuned drilling mechanism drops as the frequencymoves away from the stick-slip frequency, but the damping never dropsbelow zero, meaning that the drilling mechanism will never amplifytorsional vibrations of higher modes. In the second embodiment, thesystem comprises a PI or PID type drive speed controller being tuned sothat the drilling mechanism has a wider absorption bandwidth ofoscillation frequencies which includes both a fundamental mode and atleast one higher mode of stick-slip oscillations. The tuning in thesecond embodiment uses inertia compensation to reduce an effectiveinertia of the drilling mechanism as seen by the controller and therebyimprove the absorption bandwidth. An alternative to tuning the PI or PIDcontroller is to change into a higher gear on the drilling mechanism.

Embodiments of the invention are suitable for implementation in the PLCcontrolling a drilling mechanism. The tuned PI-controller can either beimplemented in the PLC itself or, alternatively, calculate the speedcontroller constants P and I and pass to the inherent digital speedcontroller of the top drive motors. Embodiments of the invention alsoinclude other useful aspects, including a screen based user interface,automatic determination of the stick-slip frequency, estimation ofinstantaneous bit speed and calculation of a stick-slip severity. Thelatter two are based on the drill string geometry and the measuredtorque signal.

In conclusion, therefore, it is seen that the embodiments of theinvention disclosed herein and those covered by the appended claims arewell adapted to carry out the objectives and obtain the ends set forth.Certain changes can be made in the subject matter without departing fromthe spirit and the scope of this disclosure. It is realized that changesare possible within the scope of this disclosure and it is furtherintended that each element or step recited in any of the followingclaims is to be understood as referring to the step literally and/or toall equivalent elements or steps. The following claims are intended tocover the disclosed principles and embodiments of the invention asbroadly as legally possible in whatever form it may be utilized. Theinvention claimed herein is new and novel in accordance with 35 U.S.C.§102 and satisfies the conditions for patentability in §102. Theinvention claimed herein is not obvious in accordance with 35 U.S.C.§103 and satisfies the conditions for patentability in §103. Thisspecification and the claims that follow are in accordance with all ofthe requirements of 35 U.S.C. §112. The inventors may rely on theDoctrine of Equivalents to determine and assess the scope of theirinvention and of the claims that follow as they may pertain to apparatusnot materially departing from, but outside of, the literal scope of theinvention as set forth in the following claims. All patents, patentapplications and scientific papers identified herein are incorporatedfully herein for all purposes.

1. A method of estimating the instantaneous rotational speed of a bottomhole assembly at the lower end of a drill string the method comprisingdriving the drill string by a drilling mechanism at the upper end of thedrill string; estimating a fundamental frequency of stick-sliposcillations suffered by the drill string; determining variations in adrive torque of said drilling mechanism, combining a known torsionalcompliance of said drill string with said variations in a drive torque,and providing an output signal representing said instantaneousrotational speed.
 2. A method according to claim 1, wherein saidvariations in drive torque are expressed only at said fundamentalfrequency, whereby said estimating the instantaneous rotational speed isimplemented by a PLC and performed in real time.
 3. A method accordingto claim 1, wherein said determining step comprises band pass filteringa drive torque signal with a band pass filter centred on saidfundamental frequency.
 4. A method according to claim 3, furthercomprising: determining a downhole speed using a total static drillstring compliance and a phase parameter, and determining the sum of (i)a low pass filtered signal representing a speed of rotation of saiddrilling mechanism and (ii) said downhole speed.
 5. A method accordingto claim 1, further comprising: determining an estimate of instantaneousrotational speed periodically; and outputting said estimate on adriller's console whereby a driller is provided with a real-timeestimate of the instantaneous rotational speed of said bottom holeassembly.
 6. A method according to claim 1, further comprising:determining a stick-slip severity as a ratio of dynamic downhole speedamplitude over a mean rotational speed of said drilling mechanism, whichstick-slip severity is useable to provide an output signal indicatingthe severity of stick-slip at that point in time.
 7. A method accordingto claim 1, further comprising: damping said stick-slip oscillationsusing said drilling mechanism; and controlling the speed of rotation ofsaid drilling mechanism using a PI controller; characterised by the stepof tuning said PI controller so that said drilling mechanism absorbsmost torsional energy from said drill string at a frequency that is ator near a fundamental frequency of said stick-slip oscillations.
 8. Amethod according to claim 7, wherein said stick-slip oscillationscomprise torsional waves propagating along said drill string, andwherein the tuning comprises adjusting an I-term of said PI controllerto be dependent on a period of said fundamental frequency of said stickslip oscillations and on an effective inertia of said drillingmechanism, whereby said drilling mechanism has a frequency dependentreflection coefficient of said torsional waves, which reflectioncoefficient is substantially at a minimum at or near said fundamentalfrequency of stick-slip oscillations.
 9. A method according to claim 8,further comprising adjusting said I-term according to I=ω_(s) ²J whereω_(s) is an estimated angular frequency of said stick-slip oscillationsand J is the effective inertia of said drilling mechanism.
 10. A methodaccording to claim 8, wherein said effective inertia comprises the totalmechanical inertia of said drilling mechanism at an output shaftthereof.
 11. A method according to claim 7, further comprising reducingan effective inertia of said drilling mechanism, whereby a dampingeffect of said drilling mechanism is increased for frequencies abovesaid fundamental frequency.
 12. A method according to claim 11, whereinthe reducing said effective inertia comprises tuning said PI controllerwith an additional torque term that is proportional to the angularacceleration of said drilling mechanism.
 13. A drilling mechanism foruse in drilling a borehole, the drilling mechanism comprising: anelectronic controller having a memory; wherein the memory storescomputer executable instructions that when executed cause saidelectronic controller to: estimate the instantaneous rotational seed ofa bottom hole assembly at the lower end of a drill string that, in use,is driven by said drilling mechanism at the upper end of said drillstring, and which drill string is suspected of suffering stick-sliposcillations that have an estimated or observed fundamental frequency;determine variations in a drive torque of said drilling mechanism;combine a known torsional compliance of said drill string with saidvariations in a drive torque; and provide an output signal representingsaid instantaneous rotational speed.
 14. An electronic controller foruse with a drilling mechanism for drilling a borehole, the electroniccontroller comprising: a memory; and an optional PI controller; whereinthe memory stores computer executable instructions that when executedcause said electronic controller to: estimate the instantaneousrotational steed of a bottom hole assembly at the lower end of a drillstring that, in use, is driven by said drilling mechanism at the upperend of said drill string, and which drill string is suspected ofsuffering stick-slip oscillations that have an estimated or observedfundamental frequency; determine variations in a drive torque of saiddrilling mechanism; combine a known torsional compliance of said drillstring with said variations in a drive torque; and provide an outputsignal representing said instantaneous rotational speed.
 15. A method ofupgrading a drilling mechanism on a drilling rig, the method comprising:uploading computer executable instructions to an electronic controlleron said drilling rig, which electronic controller is for controlling ormonitoring operation of said drilling mechanism, wherein said computerexecutable instructions comprise instructions for causing the electroniccontroller to: estimate the instantaneous rotational speed of a bottomhole assembly at the lower end of a drill string that, in use, is drivenby said drilling mechanism at the upper end of said drill string, andwhich drill string is suspected of suffering stick-slip oscillationsthat have an estimated or observed fundamental frequency; determinevariations in a drive torque of said drilling mechanism; combine a knowntorsional compliance of said drill string with said variations in adrive torque; and provide an output signal representing saidinstantaneous rotational speed.
 16. A drilling mechanism according toclaim 13, wherein the executable instructions cause the electroniccontroller to express said variations in drive torque only at saidfundamental frequency, and to thereby estimate the instantaneousrotational speed in real time.
 17. A drilling mechanism according toclaim 13, wherein the executable instructions cause the electroniccontroller to band pass filter a drive torque signal with a band passfilter centred on said fundamental frequency.
 18. A drilling mechanismaccording to claim 13, wherein the executable instructions cause theelectronic controller to: determine an estimate of instantaneousrotational speed periodically; and output said estimate on a driller'sconsole whereby a driller is provided with a real-time estimate of theinstantaneous rotational speed of said bottom hole assembly.
 19. Adrilling mechanism according to claim 13, wherein the executableinstructions cause the electronic controller to determine a stick-slipseverity as a ratio of dynamic downhole speed amplitude over a meanrotational speed of said drilling mechanism, which stick-slip severityis useable to provide an output signal indicating the severity ofstick-slip at that point in time.
 20. A drilling mechanism according toclaim 13, wherein the executable instructions cause the electroniccontroller to: damp said stick-slip oscillations using said drillingmechanism; and control the speed of rotation of said drilling mechanismusing a PI controller; the control characterized by: tuning said PIcontroller so that said drilling mechanism absorbs most torsional energyfrom said drill string at a frequency that is at or near a fundamentalfrequency of said stick-slip oscillations.